Title: CVaR Approximations for Minimax and Robust Convex Optimization
Speaker: Prof. Huifu Xu
Affiliation: School of Engineering and Mathematical Sciences at City University of London
Location: Room 218 Huxley Building
Time: 2:00pm

Abstract. Conditional value at risk (CVaR) has been widely used as a risk measure in finance. In this work, we propose to randomize decision variables of a deterministic parametric maximization problem and approximate the optimal (maximum) value by the CVaR of the randomized function. One of the main advantages of this approach is that computing CVaR is down to solving a one dimensional convex optimization problem even when the original problem is multi-dimensional and nonconvex. We apply the approximation scheme to a minimax (robust) optimization problem and a convex optimization problem with semi-infinite constraints indexed by uncertain parameters. In the latter application we use CVaR to approximate the semi-infinite constraint and then apply the Monte Carlo sampling method to discretize the CVaR approximated constraint. This approach is closely related to a popular randomization approach proposed by Calafiore and Campi where the continuum of uncertainty parameters is discretized through sampling. Error bounds for the optimal solutions of the approximate problems are derived under some moderate conditions and some numerical test results are reported. The proposed methods can be applied to distributional optimization where the distribution set is constructed through moments.

About the speaker. Huifu Xu is a Professor of Operational Research in the School of Engineering and Mathematical Sciences at City University of London. Before joining City University, he was a Senior Lecturer of Operational Research in the School of Mathematics at the University of Southampton. His expertise is in continuous optimization and operational research, including developing numerical methods and underlying theory for continuous optimization problems, particularly those involving uncertain data and/or equilibrium constraints. Over the past ten years, he has been actively working on stochastic mathematical programs with equilibrium problems and is recently developing interest in robust approaches for stochastic optimization and equilibrium problems. Huifu has published about 60 papers most of which are in the international journals of optimization and operational research.